Abstract
We shall try to defend two non-standard views that run counter to two well-entrenched familiar views. The standard views are (1) the universal and existential quantifiers of first-order logic are not modal operators, and (2) the quantifiers are extensional. If that is correct then the counterclaims create genuine problems for some traditional philosophical doctrines.
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Notes
As a set of ordered pairs of worlds of the model.
We take an implication relation to be any relation satisfying Gentzen’s structural conditions.
Unless specified otherwise, we consider here only those implication structures with classical negation.
Hereafter, when I use “modal operator” I shall mean “structural modal operator”.
Unless indicated otherwise, later uses of “contingent” are understood as modally contingent, using the possibility modal of the structure.
Here we have expressed the contingency condition proof-theoretically rather than semantically, which would require only that \(\diamond (\hbox {Ao}) \wedge \diamond (\lnot \hbox {Ao})\) be true.
“Contingency” should not be confused with “empirical”. As we shall see below (§ 4.c), for the Gödel-Löb modal on Peano Arithmetic, the Gödel sentence G is contingent.
In the simplest case for predicates Px, and Qx, the first predicate implies the second if and only if for all a in O, P(a) implies Q(a), where the implication relation is that used for the sentences of S. The more general case when predicates and relations of different adicity are involved, can be covered by using the Tarski (1983)– Mostowski (1966) treatment of predicates with finite bases as mappings of infinite sequences of elements of O into the set S. For details see Koslow (1992).
We shall show below (§ 5) that this condition is equivalent to the modal condition that there is a contingent predicate Px in the extended implication structure -i.e one such that \((\exists \hbox {x})(\hbox {Px}) \wedge (\exists \hbox {x})(\lnot \hbox {Px})\) is not a contradiction.
Several caveats. (1) The double-headed arrow is meant to be the material conditional, (2) no semantic notions such as “true” and “extension” are used in this expression of extensionality. The biconditional which we use, whether between sentences or between predicates is explained by using the familiar Gentzen introduction and eliminations conditions for the biconditional.
There is a proof that any structural modal on any classical implication structure is non-extensional, when the extensionality of an operator \(\upvarphi \) is characterized semantically, so that if A and B have the same truth value under a normal truth-value assignment so too do \(\upvarphi \)(A), and \(\upvarphi \)(B). Cf. (Koslow (1992), p. 263). Here we use a different proof of that fact, using the assumption that the implication structure has a modally contingent element in it.
Here the implication relation is to be understood as a deduction involving the axioms for probability theory.
These axioms for probability theory (K1) and K(2) are due to Kolmogorov who stated them in a schematic way that allowed probability theory to cover the special case of any field of sets whose members he called events, but which are sets. I believe that a parallel argument for non-extensionality can also be given for probabilities of events: A sketch: let S be a set of sets which Kolmogorov (1956) required to be a field, and the subsets of S are what he terms “events” For any two such events, e and f, we shall say that they are concomitant,\((\hbox {e} \approx \hbox {f})\) , if and only if [e occurs if and only if f occurs]. One of the Kolmogorov axioms will be that P(S) \(=\) 1, - that is, the probability that some event of S occurs, is one. For any event e, the negation \(\lnot \hbox {e}\) is the event S – e (the relative complement of e in S (i.e. the subset of S consisting of all the members of S, excluding the set e). We can then express the idea that probability for events is extensional this way: For any events e and f, If Ext(P), then \((\hbox {e} \approx \hbox {f})\) implies that P(e occurs) \(=\) r if and only if P(f occurs) \(=\) r, for all real numbers r in the closed interval [0, 1].
Consider the special case, for e and S. Extensionality then requires that if \(\hbox {e} \approx \hbox {S}\), then P(e occurs) \(=\) 1 if and only if P(S occurs) \(=\) 1. Now \(\hbox {e} \approx \hbox {S}\) is equivalent to e occurs (this is the analogue to the propositional case when \((\hbox {A} \leftrightarrow \Omega ) \Leftrightarrow \hbox {A}\), and P(S) \(=\) 1 is an axiom (this is the analogue to the propositional case when P(\(\Omega ) =\) 1). Therefore we have [e occurs \(\Rightarrow \) P(e occurs) \(=\) 1. We also then have [(\(\lnot \)e) occurs \(\Rightarrow \) P(e occurs) \(=\) 0]. Therefore it is a thesis that for any event e, [P(e) \(=\) 1) \(\Rightarrow \) P(e) \(=\) 0]. Consequently, if the probability of events is not trivial then it is non-extensional.
In §3, condition (4).
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Koslow, A. The modality and non-extensionality of the quantifiers. Synthese 196, 2545–2554 (2019). https://doi.org/10.1007/s11229-014-0539-6
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DOI: https://doi.org/10.1007/s11229-014-0539-6